MIP Solvers

A mixed-integer programming (MIP) solver is a type of optimization software that can solve mathematical models in which some of the decision variables are integers. MIP solvers typically use a branch-and-cut algorithm to identify the best possible solution given an objective function.

A MIP solver generally works by applying a branch-and-bound algorithm, which divides the problem into smaller subproblems. These subproblems are then solved recursively to find the best solution.

At each node of the branching tree, the solver relaxes the integrality constraints and solves a linear programming (LP) problem, which provides either a lower bound or an upper bound (for maximization and minimization problems, respectively).

The solver will then compare the LP solution with the incumbent solution and decide whether to remove unnecessary branches (if it cannot improve the current best solution) or to branch further (if it can potentially improve the current best solution).

The main difference between MIP and LP is that MIP includes both integer and continuous variables, while LP includes only continuous variables.

The addition of integer variables makes MIP problems more complex and computationally challenging to solve compared to LP problems.

MIP is widely used across a variety of fields, including:

• Supply chain management: MIP can be used to optimize logistics, inventory levels, production schedules, and more.
• Financial services: In the financial services industry, MIP is often used for portfolio optimization and risk management to improve operational efficiency, profitability, and regulatory compliance.
• Telecommunications: MIP can help telecom companies determine optimal network design and bandwidth allocation.
• Energy: In the energy sector, MIP can help companies streamline power generation and optimize distribution planning while mitigating risk in an increasingly competitive market.
• Manufacturing: By using a MIP solver to optimize production processes and schedules, manufacturers can increase efficiency and potentially reduce costs.

The key components of a MIP problem :

• Objective Function: The function to be optimized.
• Decision Variables: The variables that determine the outcome of the objective function.
• Constraints: The conditions that the solution must satisfy.
• Set of Indices: Set indexing is the feature that permits a concise model to describe a large mathematical program.

Common algorithms used in MIP solvers include:

• Branch-and-Bound: This is a tree-based method for systematically exploring feasible solutions.
• Cutting Planes: This method tightens the formulation by removing undesirable fractional solutions without creating additional sub-problems.
• Heuristics: Heuristics are approximate methods that are often used to provide initial solutions or improve existing ones.

To model a problem for a MIP solver, you need to:

1. Describe the problem in business terms: What do you need to achieve? What are your obstacles?
2. Formulate the problem as a mathematical model.
3. Translate the mathematical model into code that the solver can understand.